Igusa zeta-function

In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.

Definition

For a prime number p let K be a p-adic field, i.e. [K: \mathbb{Q}_p]<\infty , R the valuation ring and P the maximal ideal. For z \in K \operatorname{ord}(z) denotes the valuation of z, \mid z \mid = q^{-\operatorname{ord}(z)}, and ac(z)=z \pi^{-\operatorname{ord}(z)} for a uniformizing parameter π of R.

Furthermore let \phi : K^n \mapsto \mathbb{C} be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let \chi be a character of K*.

In this situation one associates to a non-constant polynomial f(x_1, \ldots, x_n) \in K[x_1,\ldots,x_n] the Igusa zeta function

Z_\phi(s,\chi) = \int_{K^n} \phi(x_1,\ldots,x_n) \chi(ac(f(x_1,\ldots,x_n))) |f(x_1,\ldots,x_n)|^s \, dx

where s \in \mathbb{C}, \operatorname{Re}(s)>0, and dx is Haar measure so normalized that R^n has measure 1.

Igusa's theorem

Jun-Ichi Igusa (1974) showed that Z_\phi (s,\chi) is a rational function in t=q^{-s}. The proof uses Heisuke Hironaka's theorem about the resolution of singularities. Later, an entirely different proof was given by Jan Denef using p-adic cell decomposition. Little is known, however, about explicit formulas. (There are some results about Igusa zeta functions of Fermat varieties.)

Congruences modulo powers of P

Henceforth we take \phi to be the characteristic function of R^n and \chi to be the trivial character. Let N_i denote the number of solutions of the congruence

f(x_1,\ldots,x_n) \equiv 0 \mod P^i.

Then the Igusa zeta function

Z(t)= \int_{R^n} |f(x_1,\ldots,x_n)|^s \, dx

is closely related to the Poincaré series

P(t)= \sum_{i=0}^{\infty} q^{-in}N_i t^i

by

P(t)= \frac{1-t Z(t)}{1-t}.

References

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